Abstract
We define a Wang tile set \(\mathcal {U}\) of cardinality 19 and show that the set \(\Omega _\mathcal {U}\) of all valid Wang tilings \(\mathbb {Z}^2\rightarrow \mathcal {U}\) is self-similar, aperiodic and is a minimal subshift of \(\mathcal {U}^{\mathbb {Z}^2}\). Thus \(\mathcal {U}\) is the second smallest self-similar aperiodic Wang tile set known after Ammann’s set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable 2-dimensional morphism \(\omega :\Omega _\mathcal {U}\rightarrow \Omega _\mathcal {U}\) that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles.
Similar content being viewed by others
References
Ammann, R., Grünbaum, B., Shephard, G.C.: Aperiodic tiles. Discrete Comput. Geom. 8(1), 1–25 (1992)
Akiyama, S.: A note on aperiodic Ammann tiles. Discrete Comput. Geom. 48(3), 702–710 (2012)
Akiyama, S., Tan, B., Yuasa, H.: On B. Mossé’s unilateral recognizability theorem (2017). arXiv:1801.03536
Berthé, V., Delecroix, V.: Beyond substitutive dynamical systems: \(S\)-adic expansions. In: Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, pp. 81–123. Research Institute for Mathematical Sciences (RIMS), Kyoto, (2014)
Berger, R.: The Undecidability of the Domino Problem. ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, Harvard University (1965)
Baake, M., Grimm, U.: Aperiodic Order. Vol. 1, Volume 149 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2013)
Berthé, V., Rigo, M. (eds.): Combinatorics, Automata and Number Theory, Volume 135 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2010)
Berthé, V., Steiner, W., Thuswaldner, J., Yassawi, R.: Recognizability for sequences of morphisms (2017). arXiv:1705.00167
Charlier, E.: Abstract Numeration Systems: Recognizability, Decidability, Multidimensional S-automatic Words, and Real Numbers. Ph.D. Thesis, Université de Liège, Liège, Belgique (2009)
Charlier, E., Kärki, T., Rigo, M.: Multidimensional generalized automatic sequences and shape-symmetric morphic words. Discrete Math. 310(6–7), 1238–1252 (2010)
Culik II, K.: An aperiodic set of \(13\) Wang tiles. Discrete Math. 160(1–3), 245–251 (1996)
Durand, F.: A characterization of substitutive sequences using return words. Discrete Math. 179(1–3), 89–101 (1998)
Frank, N.P.: Detecting combinatorial hierarchy in tilings using derived Voronoï tesselations. Discrete Comput. Geom. 29(3), 459–476 (2003)
Frank, N.P.: Introduction to hierarchical tiling dynamical systems. In: Tiling and Recurrence, December 4–8 2017, CIRM, Marseille Luminy, France. https://arxiv.org/abs/1802.09956 (2017)
Frank, N.P., Sadun, L.: Fusion: a general framework for hierarchical tilings of \(\mathbb{R}^d\). Geom. Dedicata 171, 149–186 (2014)
Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (Version 8.0.0). http://www.gurobi.com (2018)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman and Company, New York (1987)
Jeandel, E., Rao, M.: An aperiodic set of 11 Wang tiles (2015). arXiv:1506.06492
Kari, J.: A small aperiodic set of Wang tiles. Discrete Math. 160(1–3), 259–264 (1996)
Knuth, D.E.: The Art of Computer Programming. Vol. 1: Fundamental Algorithms. Second Printing. Addison-Wesley Publishing Co. (1969)
Labbé, S.: S. Labbé’s Research Code (Version 0.4.2). https://pypi.python.org/pypi/slabbe/ (2018). Accessed 25 July 2018
Labbé, S.: Substitutive structure of Jeandel–Raoaperiodic tilings (2018) (In preparation)
Mossé, B.: Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theor. Comput. Sci. 99(2), 327–334 (1992)
Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. Anal. Math. 53, 139–186 (1989)
Ollinger, N.: Two-by-two substitution systems and the undecidability of the domino problem. In: Logic and Theory of Algorithms, Volume 5028 of Lecture Notes in Computer Science, pp. 476–485. Springer, Berlin (2008)
Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971)
Sage Developers: SageMath, the Sage Mathematics Software System (Version 8.2). http://www.sagemath.org (2018). Accessed 25 July 2018
Schmidt, K.: Multi-dimensional symbolic dynamical systems. In: Codes, Systems, and Graphical Models (Minneapolis, MN, 1999), Volume 123 of IMA Volumes in Mathematics and its Applications, pp. 67–82. Springer, New York (2001)
Solomyak, B.: Dynamics of self-similar tilings. Ergodic Theory Dyn. Syst. 17(3), 695–738 (1997)
Solomyak, B.: Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2), 265–279 (1998)
Wang, H.: Proving theorems by pattern recognition—II. Bell Syst. Tech. J. 40(1), 1–41 (1961)
Acknowledgements
I want to thank Michaël Rao for the talk he made (Combinatorics on words and tilings, CRM, Montréal, April 2017) from which this work is originated. I want to thank Vincent Delecroix for many helpful discussions at LaBRI in Bordeaux during the preparation of this article. I am also thankful to Jörg Thuswaldner, Henk Bruin, for inviting me to present this work (Substitutions and tiling spaces, University of Vienna, September 2017) and to Pierre Arnoux and Shigeki Akiyama for the same reason (Tiling and Recurrence, CIRM, Marseille, December 2017). I want to thank Michael Baake for very helpful discussions on inflation rules and model sets and Shigeki Akiyama for its enthusiasm toward this project. The author is grateful to the comments of the referee which leaded to a great improvement in the presentation while reducing its size and simplifying many technical proofs into simpler ones. I also wish to thank David Renault for his careful reading of a preliminary version and his valuable comments. I acknowledge financial support from the Laboratoire International Franco-Québécois de Recherche en Combinatoire (LIRCO), the Agence Nationale de la Recherche “Dynamique des algorithmes du pgcd : une approche Algorithmique, Analytique, Arithmétique et Symbolique (Dyna3S)” (ANR-13-BS02-0003) and the Horizon 2020 European Research Infrastructure project OpenDreamKit (676541).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Labbé, S. A self-similar aperiodic set of 19 Wang tiles. Geom Dedicata 201, 81–109 (2019). https://doi.org/10.1007/s10711-018-0384-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-018-0384-8